Let $G$ be a linear algebraic group and $A=K[G]$ (K is a field of characterstic 0) be the coordinate ring of $G$. In Humphreys, the Lie algebra of $G$ is defined as the space of left invariant derivations of $A$ , i.e, $\mathfrak{L}(G) = \{\delta \in Der \hspace{1mm} A $ | $\delta \lambda_x = \lambda_x \delta $ for all $x \in G$ }
Note that $\lambda_x : K[G] \to K[G]$ sending $\hspace{1mm} f(y) \mapsto f(x^{-1}y) $.
Now, if we are given a connected simple algebraic group over $C$ (i.e , the defining polynomial equations of the underlying variety are over $C$), how do we see that its Lie algebra is simple ?
Further, given a simple Lie algebra how do we see that there exists a simply connected algebraic group $G$ with Lie algebra isomorphic to this given simple Lie algebra?
Please give reference to specific material which I can read to understand these claims. Any hints to the above claims would be highly appreciated. Thank you !
Among the possible references are Milne's lectures notes. In Proposition $4.1$ it is proved that a connected algebraic group $G$ is semisimple if and only if its Lie algebra is semisimple. Then 1. follows together with Theorem $4.5$. Theorem $4.22$ shows the claim 2, for fields of characteristic zero.